Abstract. Michele Sbacchi examines the impact of the discipline
of Euclidean geometry upon architecture and, more specifically,
upon theory of architecture. Special attention is given to the
work of Guarino Guarini, the 17th century Italian architect and
mathematician who, more than any other architect, was involved
in Euclidean geometry. Furthermore, the analysis shows how, within
the realm of architecture, a complementary opposition can be traced
between what is called "Pythagorean numerology" and
"Euclidean geometry." These two disciplines epitomized
two overlapping ways of conceiving architectural design.

INTRODUCTIONIt is well known that one of the basic
branches of geometry which, almost unchanged, we still use today
was codified by Euclid at Alexandria during the time of Ptolemy
I Soter (323-285/83 BC) in thirteen books called Stoicheia
(Elements). This overwhelmingly influential text deals
with planar geometry and contains the basic definitions of the
geometric elements such as the very famous ones of point, line
and surface: "A point is that which has no part;" "Line
is breathless length;" "A surface is that which has
length and breadth only" [Euclid 1956, I:153]. It also contains
a whole range of propositions where the features of increasingly
complex geometric figures are defined. Furthermore, Euclid provides
procedures to generate planar shapes and solids and, generally
speaking, to solve geometrical problems. Familiarity with the
Elements allows virtually anyone to master the majority
of geometrical topics. Although all this is well known I nevertheless
find it necessary, given our misleading post-Euclidean standpoint,
to underline that 'Euclidean Geometry' was 'Geometry' tout
court until the 17th century. For it wasonly from the second
half of 17th century that other branches of geometry were developednotably
analytic and projective geometry and, much later, topology. Yet
these disciplines, rather than challenging the validity of Euclidean
geometry, opened up complementary understandings; therefore they
flanked Euclid's doctrine, thus confirming its effectiveness.
In fact, Euclidean geometry is still an essential part of the
curriculum in high schools worldwide, as it was in the quadrivium
during the Middle Ages. That is not to say that Euclid's teaching
has never been questioned. In fact a long-standing tradition
does not necessarily imply a positive reverence: some Euclidean
topics have, indeed, undergone violent attacks and have fostered
huge debates. The ever-rising polemic about the postulate of
the parallels is just one notorious example of the many controversies
scattered throughout its somewhat disquieted existence.

Euclid was far from being an original writer. Although conventionally
referred to as the inventor of the discipline, he was hardly
an isolated genius. Historians of mathematics have clarified
how he drew from other sourcesmainly Theaetetus and Eudoxus.[1] Hence,
rather than inventing, he mostly systematized a corpus of knowledge
that circulated among Greek scholars in somewhat rough forms.
Therefore, Euclid's great merit lies in the exceptional ability
to illustrate and synthesize.. Although marred by contradictions
and gaps, the Elements, in its time, represented a gigantic
step forward, especially compared to the fragmentary way in which
geometry was known and transmitted. It soon became an immensely
useful text for all the fields where geometry was applied. Optics,
mensuration, surveying, navigation, astronomy, agriculture and
architecture all benefitted in various ways from a newly comprehensive
set of rules able to overcome geometrical problems. As its popularity
grew, the Elements went through several translations.
Following the destiny of most Greek scientific texts, it was
soon translated into Arabic and was known through this language
for almost fifteen centuries. A well-known Latin translation
was made by Adelard of Bath in the 12th century but at least
another translation existed earlier.[2] Campano's Latin translation of 1482
was the first to be published. Nevertheless a translation directly
from Greek into Latin was made by Bartolomeo Zamberti in 1505.
Federico Commandino's Latin edition of 1572 was to become the
standard one. The first English translation is due to Henry Billingsley
in 1570, with a preface by John Dee [Wittkower 1974:98; Rykwert
1980:123]. No less significant are the commentaries upon the
Stoicheia, if only because they witness the continuous debates
that scholars engaged in about the text. Certainly the most renowned
commentary is the one made in the 5th century A.D. by Proclus
on the First Book. Because of this vast and lasting tradition,
the Elements may be appropriately compared to the Bible
or to the Timaeus as a cornerstone of Western culture [Field
1984:291].

ARCHITECTURE THEORY, GEOMETRY AND NUMBERArchitecture, a
discipline concerned with the making of forms, perhaps profitted
most from this knowledge. I find it unnecessary to dwell here
upon such a vast and overstudied issue as the relationship between
architecture and geometry. Instead, it suffices to stress that
the geometrical understanding of, say, Vitruvius, Viollet Le
Duc and Le Corbusier was basically the Euclidean one  that
of the Elements. It is nevertheless true that the other
branches of geometry, which arose from the 17th century on, affected
architecture, but this can be considered a comparatively minor
phenomenon. In fact, the influence exerted by projective geometry
or by topology on architecture is by no means comparable to the
overwhelming use of Euclidean geometry within architectural design
throughout history.

The relevance of Euclidean methods for the making of architecture
has been recently underlined by scholars, especially as against
the predominance of the Vitruvian theory. According to these
studies [Rykwert 1985; Shelby 1977], among masons and carpenters
Euclidean procedures and, indeed, sleights of hand were quite
widespread. Although this building culture went through an oral
transmission, documents do exist from which it can be understood
that it was surely a conscious knowledge. 'Clerke Euclide' is
explicitly referred to in the few remaining manuscripts.[3] Probably
the phenomenon was much wider than what has been thought so far,
for the lack of traces has considerably belittled it. We can
believe that during the Middle Ages, to make architecture, the
Euclidean lines, easily drawn and visualized, were most often
a good alternative to more complicated numerological calculations.
Hence we can assume that an 'Euclidean culture associated with
architecture,' existed for a long time and that it was probably
the preeminent one among the masses and the workers.

Yet among the refined circles of patrons and architects the
rather different Vitruvian tradition was also in effect at the
same time [Rykwert 1985:26]. This tradition was based on the
Pythagorean-Platonic idea that proportions and numerical ratios
regulated the harmony of the world. The memorandum of Francesco
Giorgi for the church of S. Francesco della Vigna in Venice,
is probably the most eloquent example illustrating how substantial
this idea was considered to be for architecture [Moschini 1815,
I:55-56; Wittkower 1949:136ff]. This document reflects Giorgi's
Neoplatonic theories, developed broadly in his De Harmonia
mundi totius, published in Venice in 1525, which, together
with Marsilio Ficino's work, can be taken as a milestone of Neoplatonic
cabalistic mysticism. The whole theory, whose realm is of course
much wider than the mere architectural application, was built
around the notion of proportion, as Plato understood it in the
Timaeus. Furthermore, it was grounded on the analogy between
musical and visual ratios, established by Pythagoras: he maintained
that numerical ratios existed between pitches of sounds, obtained
with certain strings, and the lengths of these strings. Hence,
the belief that an underlying harmony of numbers was acting in
both music and architecture, the domain respectively of the noble
senses of hearing and of sight. In architecture numbers operated
for two different purposes: the determination of overall proportions
in buildings and the modular construction of architectural orders.
The first regarded the reciprocal dimensions of height, width
and length in rooms as well as in the building as a whole. The
second was what Vitruvius called commodulatio.[4] According to this procedure, a module
was established  generally half the diameter of the column
 from which all the dimensions of the orders could be derived.
The order determined the numerical system to adopt and, thus,
every element of the architectural order was determined by a
ratio related to the module. Indeed it was possible to express
architecture by an algorithm [Hersey 1976:24]. Simply by mentioning
the style a numerical formula was implied and the dimensions
of the order could be constructed. These two design procedures
are both clearly governed by numerical ratios  series of
numbers whose reciprocal relationships embodied the rules of
universal harmony.

If we now compare again these procedures with the Euclidean
ones, it appears more clearly that the difference between the
two systems is a significant one: according to the Vitruvian,
multiplications and subdivisions of numbers regulated architectural
shapes and dimensions; adopting Euclidean constructions, instead,
architecture and its elements were made out of lines, by means
of compass and straightedge. The 'Pythagorean theory of numbers'
and the 'Euclidean geometry of lines' established thus a polarity
within the theory of architecture.[5] Both disciplines were backed up and,
in a way, symbolized by two great texts of antiquity: the Timaeus
and the Elements.[6]
Although in architecture the dichotomy was brought about substantially
by the issue of proportion, the difference is, in fact, a more
general one. Every shape and not only proportional elements can
be determined either by the tracing of a line or by a numerical
calculation. This twofold design option is somehow implied in
the epistemological difference between geometry and arithmetic.
Socrates' remark, in Plato's Meno, to his slave who hesitated
to calculate the diagonal of the square, epitomizes the two alternatives:
"If you do not want to work out a number for it, trace it"
[Plato Meno 84].

I have outlined how, during the Middle Ages, Euclidean and
Vitruvian procedures empirically coexisted within building practice.
This situation would undergo an important change in the 17th
century. During the Renaissance the advent of an established
written architectural theory, based as it was on the dialogue
with Vitruvius' text, fostered the neo-Pythagorean numerological
aspect of architecture. Leon Battista Alberti, the most important
Renaissance architectural theorist, was well aware of Euclidean
geometry,[7]
a discipline which he dealt with in one of his minor works, the
Ludi Mathematici. Yet Alberti's orthodox position within the
Classical tradition could not allow him to challenge the primacy
of numerical ratios for the making of architecture. Therefore,
not surprisingly, Euclidean methods are left out of his De
Re Aedificatoria, where he quite decidedly states that: "
... the three principal components of that whole theory [of beauty]
into which we inquire are number (numerus), what we might
call outline (finitio) and position (collocatio)"
[Alberti 1485:164v-165]. For him numbers were still the basic
source. Accordingly, his seventh and eighth books, fundamental
ones of De Re Aedificatoria, are devoted to numerical
topics. Yet it might be speculated that his emphasis on lineamenta
(lineaments) and lines, never fully understood, could be an acknowledgement
of a building practice leaning more toward geometry than toward
numerology. With Francesco di Giorgio Martini's Trattato di
Architettura Civile e Militare, the Euclidean definitions
of line, point and parallels make their first appearance within
an architectural treatise, although in a rather unsystematic
way. Serlio, later, goes a step further: his first two books
include the standard Euclidean definitions and constructions;
yet they are intended to be the grounds more for Perspective
than for Architecture. Traces of Euclidean studies can be found
also in Leonardo: the M and I nanuscripts, the Foster, Madrid
II and Atlantic codices contain Euclidean constructions and even
the literal transcription of the first page of the Elements
[Lorber 1985:114; Veltman 1986].

GUARINO GUARINI AND EUCLIDISM

It is only with Guarino Guarini,
in the second half of the 17th century, however, that Euclidean
geometry abandons the oral realm and makes its open appearance
within a treatise. His posthumously published Architettura
Civile, written presumably between 1670 and his death, marks
a fundamental moment of the relationship between Euclidism and
theory of architecture. But first, a reflection on Guarini's
activity allows us to understand that his being the first to
include Euclidean geometry extensively within an architectural
treatise was no accident. I do not want to dwell upon his general
involvement with geometry and the vast use of geometrical schemes
for his buildings, two issues doubtlessly but loosely related
to this fact. I would rather point out more circumstantial events.
Firstly, being a professor of mathematics, Guarini was almost
unavoidably obliged to consider Euclidean geometry. His Euclidean
interests probably arose during his early teaching of Mathematics
at Messina where distinguished Euclidean scholars such as Francesco
Maurolico and his pupil Giovanni Alfonso Borelli had taught previously.
There Guarini found himself in one of the most stimulating scientific
centers of the time where a long-standing Euclidean tradition
existed.[8]
Maurolico wrote a commentary of the Elements, [9] while Borelli was author of the Euclides
Restituitus. Yet it was more likely in Paris, where Guarini
taught mathematics between 1662 and 1666, that his concern with
Euclidean geometry expanded. For there he encountered a lively
scientific milieu and particularly Francois Millet de Chales.
A most distinguished mathematician, this latter was the author
of Cursus seu mundu mathematicus, an encyclopedic work
on mathematics that also dealt with architecture.[10] More relevant to the present discussion
are Millet's two commentaries on the Elements, Les
Huit Livres d'Euclide and Les eléments d'Euclide expliqués
d'une maniere nouvelle et trés facile. Guarini was
deeply influenced by Millet [Guarini 1968:5, note 1]; he is referred
to frequently in Guarini's books, not just for geometrical or
mathematical matters. Out of this background developed Guarini's
magnum opus on geometry, the Euclides Adauctus et methodicus
mathematicaque universalis published in 1671. As the title
makes clear it, was both a commentary on the Elements
and an attempt to summarize the mathematical knowledge of the
time, much in the manner of his beloved Millet. It turned out
to be a rather successful book for it was republished five years
later. Guarini, therefore, falls well within the tradition of
Euclidean commentators. His interest for the discipline went
beyond the mere content, however, as Euclidean geometry was for
him a sort of universal key for human knowledge. The extent to
which Guarini considered Euclidean norms as the basis of every
scientific work is also clear from another work of his, the Trattato
di Fortificazione, where the Euclidean basic definitions
of point, line, etc. are provided at the very beginning as a
kind of conditional entry to the topic.[11] The same approach occurs with his Del
modo di Misurare le fabbriche, a booklet on surveying.

Architettura Civile came later; it was definitely written
after the Euclides since the latter is mentioned in it.
As I have suggested, the Euclidean intrusions in Architettura
Civile are far too many to justify them only on the grounds
of a mere unconscious professional bias. The argument that the
geometer prevailed over the architect misses the importance of
the issue. In the first treatise of the five constituting the
book, Guarini early on states his geometrical interests: "And
since Architecture, as a discipline that uses measures in every
one of its operations, depends on Geometry, and at least wants
to know its primary elements, therefore in the following chapters
we will set out those geometrical principles that are most necessary".[12] Consequently
the following chapter explores the "Principles of Geometry
necessary to Architecture." It contains the nine definitions
of point, line, surface, angle, right angle, acute angle and
parallel lines. Chapters dedicated to surfaces, rectilinear shapes,
circular shapes follow and the whole first treatise continues
basically in this way with postulates, other principles and several
typical Euclidean transformations such as "To draw a line
from a given point in order to make it touch the circle"
[Guarini 1968:41]. The Euclidean discipline of Geodesia fills
the Fifth Treatise  the way of dividing and transforming
planar shapes into other equivalents.[13] Some of these parts are literally transported
from his own Euclides, some are slightly elaborated on in light
of their architectural application. Guarini's Euclidean purismas
opposed to arithmeticsis remarkably evidenced, when, in
the Geodesia treatise, he considers progressions as purely geometrical
and not numerical [Capo 8]. The dismissal of numerical progression,
an attitude taken also by Francois Derand, was shared by those
who wanted to reestablish the foundation of logarithms from a
geometrical basis rather than from exponential equations.[14] Thus
the issue proposed is once again the opposition between the two
disciplines. In Architettura Civile, however, the most
significant fact for the purpose of my argument is that even
the theory of the orders, the very core of Vitruvian numerology,
is overshadowed by the alternative geometrical approach. Remarkably
the modular commodulatio procedure, rooted in numbers, is replaced
by a mixed system where the dimensions of the architectural elements
are determined by geometrical constructions and only in some
cases by numerical operations. Therefore, Guarini breaks away
from a long-standing tradition where the only possible way of
making the orders had to be numerical.

THE REVIVAL OF EUCLIDISMIn this revival of Euclidean culture Guarini
was not alone. His acknowledged source was the treatise of the
Milanese architect Carlo Cesare Osio. Osio's treatise, which
also bears the title Architettura Civile, sets forth a
system for the orders that is, even more geometrical than Guarini's.
Of course Osio's ideas, probably regarded as unorthodox or extravagant
by others, strongly appealed Guarini.[15] Hence, it is hardly surprising that
Osio, despite being a rather obscure architect, is taken by Guarini
as a primary authority, second only to Vitruvius, and is continuously
quoted throughout his Architettura Civile. With Guarini and Osio,
therefore, the Euclidean heritage is consciously acknowledged
within the learned realm of theory and no longer belongs to an
oral and empirical culture. Osio's Euclidean opposition to numerology
is clearly self-confessed: in the preface of his book he describes
the difficulties of the traditional modular systems: ".......such
those that (perhaps in order to avoid subdivisions that are intricate
in themselves) follow the fashion of the more modern with the
establishment of the modules, in which, relying on the discreet
property of the numbers.....".[16] And he then states that his method
will avoid the modules used by architects before him: "Thus
henceforth it always appeared that these were the possible ways,
and the only ones capable of putting in proportion the quantities
of the same order, both in themselves and amongst themselves.
And still in any case, through divine favour, I hope in this
work of mine to enrich Architecture to more certain and more
perfect effect. With Geometrical rules, which have for their
basis and support the Euclideian Demonstrations, I hope to aid...".[17] His new
attitude is also emphasized by a symbolic representation: in
the frontispiece he is significantly portrayed with two books
bearing the names of Vitruvius and Euclid, alluding unambiguously
to the double tradition I have outlined so far. Just as conscious
and deliberate is Guarini's Euclidism. Indeed Architettura
Civile turns out to be a rather peculiar trattato where Euclid
and Millet de Chalestwo geometersare advocated as
architectural authorities, even in the most quintessentially
architectural parts.[18]
The Euclideian leaning is revealed by a number of other circumstances.
In Architettura Civile quite often the elements of geometry
become the elements of architecture tout court. For Guarini,
for example, a wall is a 'surface' and a dome a 'semisphere.'
Consequently, 'architectural design' most often seems to be identified
with 'architectural drawing': as a true geometer Guarini describes
the production of the project rather than the production of the
building. In contrast to the two treatises of his pupil Vittone,
where technical problems are preeminent, Guarini's Architettura
Civile completely disregards the constructional aspect of
architecture in favor of detailed descriptions of drawing techniques.
This is striking, especially if we think of the technological
emphasis often displayed in Guarini's buildings. In this regard
it is curious that drawing tools are in fact grouped under the
title "Architectural Instruments". The problem, for
him, was not 'how to build' but 'how to draw.' Therefore, not
only Euclidean geometry has become a part of architectural theory
but it has also carried with it its implied linearis essentia
(linear-like essence) which in Guarini and Osio pervades the
all matter.

The expression linearis essentia is Francesco Barozzi's. An
outstanding mathematician and friend of Daniele Barbaro, Barozzi
was the leader of a movement of general reappraisal of Euclidean
geometry, which centered around Barozzi in Venice and Padua and
around Federico Commandino in Urbino.[19] The achievements of this group of scholars
are essential to understanding how Euclidean geometry passed
from Serlio's timid acknowledgement to Guarini's broad inclusion
within architecture.[20]
Barozzi, Barbaro, Commandino and their circles contributed to
the recognition of geometry as a modern science. Consequently
they took the rigorous rereading of the Euclidean text as a conditional
starting point. Commandino dedicated all his life to retranslating
and clarifying Greek texts on science, among them the Elements.
Franceso Barozzi edited a renowned edition of Proclus's commentary,
in which, as already noted, he acutely observed and stressed
the fundamental linear-like essence of geometry. But Barozzi
and Barbaro's epistemological interest dwelled upon another important
notion, that of "demonstration" (demonstrazione),
not coincidentally a basic requisite of the Euclidean axiomatic-deductive
procedure. For them, but also for other mathematicians of the
Paduan circle such as Giuseppe Moleto as well, the theory (teorica)
would have been valid only in conjunction with demonstrations
[Tafuri 1985:202].[21]
Barozzi also polemized with Alessandro Piccolomini and Pietro
Catena, who argued for the separation of Aristotelian syllogism
from mathematical logic, thereby putting the latter on an inferior
level. On the other hand, Barozzi in his Opusculum: in quo
una Oratio e duo Questiones, altera de Certitude et altera de
Medietate Mathematicarum continentur, dedicated to Daniele
Barbaro, stressed that "the certitude of mathematics is
contained in the syntactic rigor of demonstrations" [Tafuri
1985:206]. To carry this idea into architectural theory was,
as is well known, Barbaro's task in his Vitruvian commentary,
where syllogism (for Barbaro, discorso) and demonstration
are key elements. Therefore not only was geometry at that time
compellingly reevaluated but the epistemological value of the
geometrical demonstration was appreciated as well, with an interesting
architectural twist.

THE DECLINE OF 17TH CENTURY PYTHAGOREAN
NUMEROLOGYIf
the general rise of geometry can explain Guarini's achievement,
another phenomenon must be considered. Guarini's Euclidism can
also be rightly inserted in a general decline of Pythagorean
numerology in the 17th century. In the fields of astronomy and
music, at that time, Kepler made an even more radical dismissal
of numerology on the grounds of the Euclidean argument. Astronomy
had been saturated with Pythagorean ideas but the Copernican
revolution shook the whole field, promoting new interpretations.
With the moon no longer considered a planet but a satellite,
Copernicus's planets became six instead of the Ptolemaic seven.
The astronomer Rheticus tried to confer meaning to this number
according to a Pythagorean understanding:

For the number six is honoured above all the others in the
sacred prophecies of God and by the Pythagoreans and the other
philosophers. What is more agreeable to God's handiwork than
this first and most perfect work should be summed up in this
first and most perfect number? [Field 1984:273]

To this Kepler replied in the Mysterium Cosmographicum
on a geometrical basis. For him the orbs were six because they
defined the spaces between the five regular solids. To substantiate
the fact that the bodies were five Kepler cited the last proposition
of Book XIII of Euclid's Elements. This should not be
considered coincidental for, indeed, Euclid was held in the highest
consideration by Kepler: for example, in a letter to Heydon in
1605, he writes that the archetype of the world "lies in
Geometry, and specifically in the work of Euclid, the thrice-greatest
philosopher [et nominatim in Euclide philosopho ter maximo]"
[Field 1984:283]. But Kepler's most evident Euclidean concern
came out in the field of music, where he tried to fight the Pythagorean
conception, exactly in the realm where it was strongest. Kepler's
Harmonices Mundi is specially devoted to the founding
of musical ratios on geometry. The first book, in which Kepler
outlines his theory, is entirely devoted to geometry, the second
on music. He declares:

Since today, to judge by the books that are published, there
is a total neglect of the intellectual distinctions to be made
among geometrical entities, I thought fit to state at the outset
that it is from the divisions of the circle into equal aliquot
parts, by means of geometrical constructions [i.e., using straight
edge and compasses], that is, from the constructible Regular
plane figures, that we should seek the causes of Harmonic proportions.[Field
1984:283]

Judith Field has pointed out that "... the weight of
the geometrical work in Harmonices Mundi ... must be seen
as indicating that he took very seriously his endeavor to prove
that God was a Platonic geometer rather than a Pythagorean numerologist"
[Field 1984:284]. The case of Kepler further proves that the
opposition between Pythagorean theories and Euclidism was a vast
phenomenon which transcended the realm of architectural theory.
Moreover, Kepler's attitude reveals that the issue, far from
involving merely practical procedures, had ontological facets
in the deepest sense.

THE CONFLICT BETWEEN EUCLIDISM AND PYTHAGOREAN
NUMEROLOGYTo
complete my analysis I shall lastly consider a fundamental antithesis.
In fact, the conflict between Euclidism and Pythagorean numerology
is mirrored by the analogous dualism between two opposite ways
of conceiving quantities, as continuous or as discrete. This
topic requires a discussion which is too vast for this essay,[22] yet a
short treatment is indispensable for the purpose of my argument.
Quantities can be intended either as the summation of infinitesimal
partshence they are discreteor as the product of
the flow of some primary entitieshence they are continuous.
This double conception goes back at least to Aristotle and has
been widely discussed over centuries. The root of the different
approach towards reality adopted in the two disciplines of geometry
and arithmetic must be sought in this very duality. In arithmetic
quantity is conceived as discrete; this means that it is represented
by entities such as numbers. This conception is grounded on two
assumptions: that things are separable and that, consequently,
they can be enumerated. The idea of quantity as discrete is therefore
an essential one for the very nature of arithmetic. The Pythagoreans'
enthusiasm about numbers celebrated mystically this very possibility.

In geometry the approach is totally different: the entities
adoptedline, volume, etc.are thought of as continuous;
they match the continuity of reality in a more comprehensive
way than the discrete ones do. For example the geometrical linenot
coincidentally taken as the symbol of the "continuous"represents
mensurable as well as incommensurable quantities, by means of
the infinite series of his points. As a matter of fact the argument
about discrete and continuous quantity has historically often
been used to distinguish geometry from arithmetic, and sometimes
to support the superiority of one over the other.[23] Geometry, in fact, often became synonymous
with continuous. Mathematicians such as Barozzi, Tartaglia or
Vivianijust to quote those from the period with which I
have mainly dealtwere well aware of this distinction, as
scientists are today. Architects, instead, only vaguely considered
it. The very learned Scamozzi and the rather minor figure Osio
are two of the few who included this topic, although very briefly,
in their treatises. Guarini, who as a mathematician and philosopher
discusses at length quantitas, continua and quantitas discreta
in his books, disregards it almost completely in his architectural
treatise.[24]
This is rather surprising because, as I have tried to demonstrate,
the field of architecture was a crucial battleground for the
two conceptions. Indeed in the making of architectural forms
the choice between a line to tracei.e. the geometical approachor
a number to calculatei.e. the numerological approachnot
only implies rather different design methods but also brings
about diverse results.

The opposition of the continuous to the discrete enlightens
how deep, conceptually, was the opposition of geometry to arithmetic.
The change that occurred in architecture at the end of the 17th
century, which witnessed a dismissal of Pythagorean numerology
in favour of a more explicit adherence to geometry, is therefore
a meaningful phenomenon. It consisted in making official rather
widespread but disguised procedures. Furthermore, its belonging
to a vast cultural phenomenonof which I have analyzed the
revival of Euclidean geometry within Italian scientific circles
and Kepler's approach in the fields of astronomy and musicfurther
magnifies its importance.

NOTES[1]In particular the whole
theory of proportionals, including the much-debated Definition
V was taken from Eudoxus of Cnido (IV c. B. C.) [Euclid 1956,
I:1]. See also [Cambiano 1967]. return
to text

[2] Heath has pointed out that a Latin translation,
earlier than Adelard's, must have been the common source for
at least three documents: Boethius, a passage in the Gromatici
and the Regius Manuscript in the King's Library of the British
Museum [Euclid 1956, I: 91-95]. return
to text

[3] Two manuscripts are located in the King's Library
of the British Museum, the Regius manuscript and the Coke manuscript.
See [Knoop 1938; Euclid 1956, I: 95; Halliwell: Rara Mathematica].
return to text

[5] Girolamo Cardano stigmatizes this opposition when
in his De subtilitate contrapposes an "Euclidis Laus,"
which praises Euclid's "inconcussa dogmatum firmitas,"
with a rather critical "Vitruvij Laus," where Vitruvius
is accused of being only a compiler. See [Oechslin 1983:23].return to text

[9] Unpublished manuscript at the Biblioteque Nationale,
Paris. He also translated Euclid's Phenomena. return
to text

[10] On Millet de Chales and 17th century encyclopedism
see [Vasoli 1978]. return to text

[11] "The Elements of Euclid are so necessary to
every science and also to whoever would advance themselves
in the military arts must believe them to be the basis, principle
and fundamental element on which to build, and beyond which to
advance, and on which to lay every speculation" ("Gli
Elementi di Euclide sono si necessari ad ogni scienza ... e pertanto
qualunque vuole avanzarsi nell'arte militare, deve credere, che
questa sia la base, il principio & il primo elemento, di
cui si compone, e sopra a cui s'avanza, e cresce ogni sua speculazione")
[Guarini 1968: 10]. return to text

[13] There were, in fact, two tradition for Geodesy.
The first referred to the lost treatise by Euclid on The Division
of Figures, of which existed an Arabic copy by Muhammed ibn Muhammed
al Bagdadi, translated into Italian in 1570. The second referred
to the Metrics of Hero. See [Guarini 1968: 389, n. 1]. return to text

[18] See [Guarini I,1] where Millet is strikingly quoted
together with Vitruvius for the definition of architecture; and
I, III, Osservazione 6, where Millet is quoted for the matter
of the respect of ancients' rules; see also III, 17, 2, where
the topic is the Doric order. return
to text

[19] Daniele Barbaro is quoted together with Vettor Fausto
and Nicoló Tartaglia as a restorer of the antique scientific
rigor in the dedication of Guidobaldo del Monte, Mechanicorum
Liber (Pesaro, 1577), quoted in [Tafuri 1985:203]. return to text

[20] To this might be added John Dee's inclusion of architecture
among the mathematical arts. return
to text

[21] The connection between syllogism and geometrical
reasoning was known since Socrates' times. See [Mueller: 292ff].
return to text

[22] A good summary is given by [Evans 1957]. See also
[Manin 1982]. return to text

[23] A position like that of Ramus is to this respect
symptomatic. On Ramus and French anti-Euclidism see [Bruyere
1984]. return to text

[24] Guarini gives this topic primary importance. His
Euclides begins with Tractatus I - De quantitate continua
and Tractatus II - De quantitate discreta; these topics
are treated also in several other parts of the book. In Placita
Philosophica one chapter deals with Quantitas and
another with De continui compositione. return
to text

Alberti, Leon Battista. 1991. On the Art
of Building in Ten Books. Neil Leach and Robert Tavernor,
trans. Cambridge, MA: MIT Press). (English translation.) To order this book from Amazon.com,
click here.

Clagett, M. 1974. The Works of F. Maurolico.
In Physis XVI, 2: 149-198.

Daye, John (John Dee). 1570. The elements
of Geometrie...of Euclide of Megara...translated into English
Toung by H.Billingsley...with a very Fruitfull Preface made by
M.J.Dee Specyfying the Chief Mathematicall Sciences, What They
are, and Whereunto Commodius. London.

ABOUT
THE AUTHORMichele Sbacchiis a researcher at the Faculty of Architecture in Palermo
where he teaches Architectural Design. He received his Master
in Architecture at Cambridge University under the supervision
of Joseph Rykwert. From 1988 until 1991 he worked as research
assistant of Rykwert at the Faculty of Architecture, University
of Pennsylvania in Philadelphia. In 1994 took his Dottorato di
Ricerca at the University of Naples and did a year's post-doctoral
work at Palermo University. He has been awarded 2nd prize at
the International Competition for Schools of architecture of
the 4a International Bienal de Sao Paulo in Brasil, 3rd prize
and special mention at the International Competition Living as
students, Bologna, and 1st price at the National Competition
for the renewal of Palermo's circular freeway. His paper "Elements"
has been selected for the conference Research by Design, Technical
University, Delft. He practises as an architect in his own office
in Palermo.